This has been annoying me for the past two days.
Suppose we have an algebra $A$ over some field and a subalgebra $B \subseteq A$. Suppose we also have that $A$ is a finitely-generated free module over $B$, so as a $B$-module $A =Bx_1 \oplus ... \oplus Bx_n$ for some $x_1,...,x_n \in A$.
Can I assume $x_1=1$?
Or is it possible there is some sort of example where although $A$ is just $n$ copies of $B$ as a module, the only way it contains $B$ as a subalgebra is a different one where the structure is more complicated and it doesn't decompose into that direct sum?